Question: All of the 5th grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$4.50$ each for students, and the group paid $$61.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$16.50$ each for teachers and $$8.00$ each for students, and the group paid $$129.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+4.5y = 61.5}$ ${16.5x+8y = 129.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-13.5y = -184.5}$ ${16.5x+8y = 129.5}$ Add the top and bottom equations together. $ -5.5y = -55 $ $ y = \dfrac{-55}{-5.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {5.5x+4.5y = 61.5}$ to find $x$ ${5.5x + 4.5}{(10)}{= 61.5}$ $5.5x+45 = 61.5$ $5.5x = 16.5$ $x = \dfrac{16.5}{5.5}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {16.5x+8y = 129.5}$ and get the same answer for $x$ ${16.5x + 8}{(10)}{= 129.5}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.